No more PM in here, the battle is still a turn based system but with a AP (Action Point) system.

There is six icon in the top-right of this screen.
These are AP. With them you can perform more than one action in one turn. Each action use one or more AP,. (Magic for example)
Each character have 3 AP and if they use 3 AP in a turn they won't be able to perform another action until the next turn.
Don't worry, if you have a skills who requiere more than 3 AP you can always borrow AP from your party members.
Skills:

Most of skills are in weapon and need SP (Skill Points) to be able to get and use them permanently.
You can obtain SP by defeating monsters and when you lvl up.
Quest:
To do quest, you need to find a NPC with a bubble like this:

He'll explain his problem and a quest will be add to your quest journal.

The numbers of stars represent the quest difficulty.
You can access to your quest journal in the menu.